For my cladistic analysis, i lay out everyone else's matrix and then
toss those characters that, somehow, i determine to be most
unrelable.

Don't.

There's no reason to throw out any characters that aren't
parsimony-uninformative (and even those don't actually hurt); judgments
of which characters are "reliable" are inevitably subjective to a large
extent. But do make sure to merge correlated characters (easier said
than done, but try).

While you're at it, look for mistakes. Any big matrix inevitably
contains typos and other glitches, and every one of them has an
unpredictable influence on the results.